For a simple compressible substance in equilibrium, the first and second laws combine into differential property relations.

In specific form,

$$T,ds=du+p,dv,$$

and, using $h=u+pv$,

$$T,ds=dh-v,dp.$$

These are often called Gibbs relations or $Tds$ relations. They connect energy and entropy properties with pressure, volume and temperature.

They are relations among neighboring equilibrium states; they do not require the actual process connecting those states to be reversible. Reversibility is needed only if one additionally identifies $Tds$ with actual heat transfer $\delta q$.

For an incompressible substance with $dv\approx0$ and $du\approx c,dT$,

$$ds\approx c\frac{dT}{T},$$

so between two states with constant $c$,

$$s_2-s_1\approx c\ln\frac{T_2}{T_1}.$$

The Gibbs relations are the main bridge from equations of state and energy properties to entropy changes.